Jean-Paul Berrut

Barycentric Lagrange Interpolation

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

Exponential convergence of a linear rational interpolant between transformed Chebyshev points

In 1988 the second author presented experimentally well-conditioned linear rational functions for global interpolation. We give here arrays of nodes for which one of these interpolants converges exponentially for analytic functions.

Recent Developments in Barycentric Rational Interpolation

In 1945, W. Taylor discovered the barycentric formula for evaluating the interpolating polynomial. In 1984, W. Werner has given first consequences of the fact that the formula usually is a rational interpolant. We review some advances since the latter paper in the use of the formula for rational interpolation.

Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval

Abstract Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of n + 1 nodes another denominator of degree n , and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection. The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes. It has to be computed numerically. For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used. The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1]. We show here how to do so, and we present numerical results.

The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points

Abstract We discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for Cebysev-Gauss-Lobatto points. We propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices.

The linear rational collocation method

We introduce the collocation method based on linear rational interpolation for solving general hyperbolic problems, prove its stability and its convergence in weighted norms and give numerical examples for its use.

THE LINEAR RATIONAL PSEUDOSPECTRAL METHOD FOR BOUNDARY VALUE PROBLEMS

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezers suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.

Rational interpolation through the optimal attachment of poles to the interpolating polynomial

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markovs inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error.

Barycentric Formulae for Cardinal (SINC-).Interpolants

SummaryWe present a barycentric representation of cardinal interpolants, as well as a weighted barycentric formula for their efficient evaluation. We also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and, in other cases, is even more accurate.In numerical examples, we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work.

Recent advances in linear barycentric rational interpolation

Well-conditioned, stable and infinitely smooth interpolation in arbitrary nodes is by no means a trivial task, even in the univariate setting considered here; already the most important case, equispaced points, is not obvious. Certain approaches have nevertheless experienced significant developments in the last decades. In this paper we review one of them, linear barycentric rational interpolation, as well as some of its applications.